362 CHAPTER 5 Exponential and Logarithmic Functions Exploration On the same viewing rectangle, graph = + = + − − Y e Y e 500 1 24 and 500 1 24 t t 1 0.03 2 0.08 What effect does the growth rate b have on the logistic growth function? Wood Products The EFISCEN wood product model classifies wood products according to their life-span. There are four classifications: short (1 year), medium short (4 years), medium long (16 years), and long (50 years). Based on data obtained from the European Forest Institute, the percentage of remaining wood products after t years for wood products with long life spans (such as those used in the building industry) is given by ( ) = + P t e 100.3952 1 0.0316 t 0.0581 (a) What is the decay rate? (b) What is the percentage of remaining wood products after 10 years? (c) How long does it take for the percentage of remaining wood products to reach 50%? (d) Explain why the numerator given in the model is reasonable. Solution EXAMPLE 6 (a) The decay rate is = − = b 0.0581 5.81% per year. (b) Evaluate ( ) P 10 . ( ) = + ≈ ⋅ P e 10 100.3952 1 0.0316 95.0 0.0581 10 So 95% of long-life-span wood products remain after 10 years. ( ) ≈ ≈ − ≈ −e t t 0.0049 ln 0.0049 0.37 14.4 days t 0.37 Divide both sides by 56.5. Rewrite as a logarithmic expression. Divide both sides by −0.37. It will take approximately 14.4 days (14 days, 10 hours) for the population to reach 180 fruit flies. (e) One-half of the carrying capacity is 115 fruit flies. Solve ( ) = P t 115 by graphing = + − y e 230 1 56.5 t 1 0.37 and = y 115, 2 and using Desmos. See Figure 59. The population will reach one-half of the carrying capacity in about 10.9 days (10 days, 22 hours). Figure 59 Desmos Look at Figure 59. Notice the point where the graph reaches 115 fruit flies (one-half of the carrying capacity): The graph changes from being concave up to being concave down. Using the language of calculus, we say the graph changes from increasing at an increasing rate to increasing at a decreasing rate. For any logistic growth function, when the population reaches one-half the carrying capacity, the population growth starts to slow down. Now Work PROBLEM 23
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